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Logarithms of Negative and Imaginary Numbers

By Euler's identity, $ e^{j\pi} = -1$ , so that

$\displaystyle \ln(-1) = j\pi
$

from which it follows that for any $ x<0$ , $ \ln(x) = j\pi + \ln(\vert x\vert)$ .

Similarly, $ e^{j\pi/2} = j$ , so that

$\displaystyle \ln(j) = j\frac{\pi}{2}
$

and for any imaginary number $ z = jy$ , $ \ln(z) = j\pi/2 + \ln(y)$ , where $ y$ is real.

Finally, from the polar representation $ z=r e^{j\theta}$ for complex numbers,

$\displaystyle \ln(z) \isdef \ln(r e^{j\theta}) = \ln(r) + j\theta
$

where $ r>0$ and $ \theta$ are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term $ e^{j\theta }$ extracts its phase (times $ j$ ).


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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